Gradients, Curls, and Divergence

Question

I got that d) does not make sense since (F dot G) yields a scalar and the gradient acts on a vector to yield another vector in space. So you would not be able to perform the gradient(F dot G)

Is that correct?

Answer 1

No, (d) is a well-defined operation because the gradient applied to a scalar function yields a vector-valued function. You want (e). If $F,G:\mathbb{R}^{3}\to\mathbb{R}^{3}$, then the operation $\nabla (F+G)$ is not well-defined.